On the singular spectrum for adiabatic quasi-periodic Schr\"odinger Operators

TL;DR

This paper investigates the spectral characteristics of quasi-periodic Schrödinger operators in the adiabatic limit, revealing conditions under which the spectrum is purely singular and providing asymptotic formulas for the Lyapunov exponent.

Contribution

It extends previous work by deriving asymptotic formulas and establishing singular spectrum properties for a broader class of adiabatic quasi-periodic Schrödinger operators.

Findings

Spectrum is purely singular in certain energy intervals.

Provides asymptotic formula for the Lyapunov exponent.

Confirms conjecture for a broader class of operators.

Abstract

In this paper we study spectral properties of a family of quasi-periodic Schr\"odinger operators on the real line in the adiabatic limit. We assume that the adiabatic iso-energetic curve has a real branch that is extended along the momentum direction. In the energy intervals where this happens, we obtain an asymptotic formula for the Lyapunov exponent and show that the spectrum is purely singular. This result was conjectured and proved in a particular case by Fedotov and Klopp in \cite{FEKL1}.